From: Wolfgang Bangerth Date: Thu, 9 Apr 2009 05:30:37 +0000 (+0000) Subject: Finish Bernoulli's law. X-Git-Tag: v8.0.0~7888 X-Git-Url: https://gitweb.dealii.org/cgi-bin/gitweb.cgi?a=commitdiff_plain;h=f069fa5632136d693838a67e17c4442635512674;p=dealii.git Finish Bernoulli's law. git-svn-id: https://svn.dealii.org/trunk@18576 0785d39b-7218-0410-832d-ea1e28bc413d --- diff --git a/deal.II/examples/step-34/doc/intro.dox b/deal.II/examples/step-34/doc/intro.dox index cfb2ffe454..98822140a1 100644 --- a/deal.II/examples/step-34/doc/intro.dox +++ b/deal.II/examples/step-34/doc/intro.dox @@ -336,7 +336,77 @@ can merge the first and third term into one: \partial_j [\partial_i (\partial_j\phi) \phi] @f} -FINISH (use that velocity is rotation-free) FINISH +We now only need to massage that last term a bit more. Using the product rule, +we get +@f{align*} + \partial_j [\partial_i (\partial_j\phi) \phi] + &= + \partial_i [\partial_j \partial_j\phi] \phi + + + \partial_i [partial_j \phi] (\partial_j \phi). +@f} +The first of these terms is zero (because, again, the summation over $j$ gives +$\Delta\phi$, which is zero). The last term can be written as $\frac 12 +\partial_i [(\partial_j\phi)(\partial_j\phi)]$ which is in the desired gradient +form. As a consequence, we can now finally state that +@f{align*} + [\mathbf{v}\cdot\nabla\mathbf{v}]_i + &= + \partial_i (\partial_j [(\partial_j\phi) \phi + v_{\infty,j} \partial_j \phi]) + - + \partial_j [\partial_i (\partial_j\phi) \phi] + \\ + &= + \partial_i + \left[ + (\partial_j\phi)(\partial_j \phi) + v_{\infty,j} \partial_j \phi + - + \frac 12 (\partial_j\phi)(\partial_j\phi) + \right], + \\ + &= + \partial_i + \left[ + \frac 12 (\partial_j\phi)(\partial_j \phi) + v_{\infty,j} \partial_j \phi + \right], +@f} +or in vector form: +@f[ + \mathbf{v}\cdot\nabla\mathbf{v} + = + \nabla + \left[ + \frac 12 \mathbf{\tilde v}^2 + + \mathbf{v}_{\infty} \cdot \mathbf{\tilde v} + \right], +@f] +or in other words: +@f[ + p + = + -\rho + \left[ + \frac 12 \mathbf{\tilde v}^2 + + \mathbf{v}_{\infty} \cdot \mathbf{\tilde v} + \right] + = + -\rho + \left[ + \frac 12 \mathbf{v}^2 + - + \frac 12 \mathbf{v}_{\infty}^2 + \right] + . +@f] +Because the pressure is only determined up to a constant (it appears only with +a gradient in the equations), an equally valid definition is +@f[ + p + = + -\frac 12 \rho \mathbf{v}^2 + . +@f] +This is exactly Bernoulli's law mentioned above.

The numerical approximation